As usual, \(\omega (G),\chi (G),\alpha (G)\)  denote the clique number, chromatic number, and independence number of a graph \(G\) , respectively. A classical result of Asplund and Grünbaum [1]} states that if \(G\)  is the intersection graph of rectangles, then \(\chi (G)\le O(\omega (G)^2)\) . After 60 years, this was only recently improved by Chalermsook and Walczak [2]} to \(\chi (G)\le O(\omega (G)\log \omega (G))\) . On the other hand, there is no known construction beating the trivial bound \(\chi (G)\ge \Omega (\omega (G))\) .
